Mathematical Modeling of Bioassays

ISSN 0006-2979, Biochemistry (Moscow), 2017, Vol. 82, No. 13, pp. 1744-1766. © Pleiades Publishing, Ltd., 2017. Original Russian Text © D. V. Sotnikov, A. V. Zherdev, B. B. Dzantiev, 2017, published in Uspekhi Biologicheskoi Khimii, 2017, Vol. 57, pp. 385-438.

REVIEW

D. V. Sotnikov, A. V. Zherdev, and B. B. Dzantiev*

Bach Institute of Biochemistry, Research Center for Biotechnology, Russian Academy of Sciences, 119071 Moscow, Russia; E-mail: [email protected], [email protected] Received September 11, 2017 Revision received September 19, 2017

Abstract—The high affinity and specificity of biological receptors determine the demand for and the intensive development of analytical systems based on use of these receptors. Therefore, theoretical concepts of the mechanisms of these systems, quantitative parameters of their reactions, and relationships between their characteristics and ligand–receptor interactions have become extremely important. Many mathematical models describing different bioassay formats have been proposed. However, there is almost no information on the comparative characteristics of these models, their assumptions, and predictive insights. In this review we suggested a set of criteria to classify various bioassays and reviewed classical and contempo- rary publications on these bioassays with special emphasis on immunochemical analysis systems as the most common and in-demand techniques. The possibilities of analytical and numerical modeling are discussed, as well as estimations of the minimum concentrations that may be detected in bioassays and recommendations for the choice of assay conditions.

DOI: 10.1134/S0006297917130119

Keywords: bioassay, immunoassay, analytical modeling, numerical modeling, theoretical detection limit

Detection methods based on the interaction of a sub- second approach is immobilization of one of the reagents stance of interest with biological receptor molecules are on the support with the possibility of further separation called bioassays. Receptors can be antibodies, enzymes, (wash-off) of the unreacted molecules, such as excess cell surface receptors, non-antibody combinatorial com- reagents, sample components, etc. Although methods pounds, oligonucleotides, peptides, lectins, etc. [1]. The that do not use labels or separation of reactants (immuno- ability of certain biological molecules to specifically rec- precipitation, immunoagglutination, immunoelec- ognize target molecules makes them an ideal tool for trophoresis, etc.) are well known and have for a long time detecting target compounds in complex mixtures. dominated the field of bioassays, currently these methods Bioassays have found broad application for routine labo- occupy a very limited niche. Since the principles of such ratory analysis in medical and veterinary diagnostics, method have been described in detail in many publica- environmental monitoring, biosafety, and many other tions [2-6], we will not discuss them in our article. areas, as well as research tools to obtain new information Mathematical modeling is an integral part of the the- on the structure and properties of various molecules and oretical basis of any assay. Development and analysis of a molecular complexes. mathematical model help to understand the mechanisms Two approaches have been found to be most promis- of processes that occur in the system, to explain various ing in the development of bioassays. The first one is incor- negative phenomena and to eliminate them. poration of various labels into the reagents with their fol- Mathematical models also have a predictive function and lowing detection in the complexes formed during the allow to evaluate the impact of various factors and param- assay. In many cases, the use of labels reduces detection eters on the assay results without long laborious experi- limits of the method by several orders of magnitude. The ments. Although any theoretical model only partially corresponds to a real process, the model reflects the general principles of the system functioning. As a rule, a model of Abbreviations: ELISA, enzyme-linked immunosorbent assay; ICA, immunochromatographic assay; PFIA, polarization fluo- an assay is considered valid if it can be used the calculate rescent immunoassay; PCR, polymerase chain reaction; RIA, the concentration of the detected complex from the ini- radioimmunoassay; RU, resonance units; SPR, surface plas- tially specified parameters, such as reagent concentra- mon resonance. tions, interaction constants, time, etc. The established * To whom correspondence should be addressed. theoretical relationship between the initial analyte con-

1744 MATHEMATICAL MODELING OF BIOASSAYS 1745 centration and the concentration of the detected complex schemes for which two options are possible: antibody describes the assay calibration curve. Based on this rela- labeling with antigen immobilization or antigen labeling tionship, it is possible to calculate the optimal reagent with antibody immobilization. ratios and the duration of the process stages, as well as In the first case (Fig. 1a), two types of antigens com- other assay parameters, e.g., detection limit, dynamic pete for the binding sites in the antibody: free antigen, the range, etc. content of which in the sample is to be measured, and Mathematical modeling of a system can be carried immobilized antigen introduced into the system in a cer- out by finding exact solution to equations that describe tain chosen amount. After interaction, antibodies bound the system in a general form (analytical modeling) or by to the immobilized antigen remain on the support, and the finding an approximate solution for specific parameter remaining antibodies (including those that reacted with values via step-by-step numerical calculations (numerical the antigen in the solution) are washed off. The higher the modeling). Modern computer technologies enable content of free antigen in the sample, the lesser amount of numerical modeling of multicomponent systems, while the label binds to the carrier and can be detected after the taking into account multiple parallel reactions, polyva- assay is completed (i.e., inverse relationship between the lent interactions, diffusion, and other processes [7-14]. analyte concentration and the recorded signal). However, non-numerical (analytical) solutions are As the interactions in the system involve two reac- preferable for understanding the functioning of assays. tions, the assay can be carried out by changing the order Although such solutions exist only for the simplest mod- of the immunoreagent interactions: (i) simultaneous els, they are actively used to develop recommendations incubation of the antibody and both types of antigen (Fig. for improving bioassay protocols [15, 16]. 2a); (ii) preincubation of the antibody and the antigen of This review focuses on the models of various bioas- interest, followed by the interaction of the antibody and says described in the literature and methods for achieving immobilized antigen (Fig. 2d), and (iii) interaction of the the optimal analytical characteristics that follow from immobilized antigen with the labeled antibody, followed these models. Various existing models and their evolution by the addition of the antigen of interest to release some are discussed. We also systemized and unified the descrip- of the bound antibodies to the solution (Fig. 2g). tions of bioassay systems proposed in other publications. By varying the duration of each stage, the range of detectable antigen concentrations can be shifted. In the second case (Fig. 1b), the assay involves inter- DIVERSITY OF BIOASSAY FORMATS actions between the immobilized antibody and two types of the antigen: labeled antigen at a known concentration The number of different biochemical assays pro- and unlabeled antigen that is present in the sample in an posed at different times is very large, and the assays are unknown (to be determined) concentration. Both types classified in different ways. The situation becomes even of antigen compete for the binding sites of immobilized more complicated because depending on the manner of antibodies. As a result, the ratio between the labeled and classification and the scope of use, the same method can unlabeled antigens in the complex with the antibody be named differently. Various bioassay formats and gener- reflects the initial ratio between these two antigen types in al principles of their development have been described in the solution. The more unlabeled antigen is present in the many reviews [17-22]. sample, the less amount of labeled antigen binds to the Let us consider one of the possible classifications of immobilized antibody (inverse relationship between the bioassay formats using immunoassay as an example. analyte concentration and the registered signal). All immunoassay methods can be subdivided based This type of assay can be also carried out in three on the structure of the assayed antigen. The first group ways: (i) simultaneous addition of the labeled and unla- consists of monovalent antigens interacting with only one beled antigens to the immobilized antibody (Fig. 2b); (ii) antibody molecule; the second group includes polyvalent interaction of the antibody with the unlabeled antigen, antigens capable of binding with several antibody mole- followed by the addition of the labeled antigen (Fig. 2e); cules. The first group is formed by low molecular weight (iii) or preincubation of the antibody with the labeled compounds: pesticides, antibiotics, peptides, etc. antigen, followed by the addition of the unlabeled antigen Biopolymers and corpuscular structures (viruses, cells) (Fig. 2h); as a result, the labeled antigen will be displaced are typical polyvalent antigens. by the unlabeled antigen from its complex with the anti- In assay systems for monovalent antigens, formation body. of the antigen–antibody complexes is not registered Let us now consider assay formats for polyvalent anti- directly1. Instead, they implement competitive assay gens. In principle, they can employ competitive assay as well; however, formation of triple, or the so-called sand- 1 Direct registration is possible, but it forfeits basic advantages wich complexes (immobilized antibody–antigen–labeled of modern assay systems: the use of labels to identify immune antibody) (Fig. 1c), provides greater sensitivity. Binding to complexes and the use of solid supports to separate them. the excess amount of immobilized antibody concentrates

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1746 SOTNIKOV et al. the antigen and provides very high sensitivity of the assay [23]. Theoretically, such system is capable of detecting even a single antigen molecule bound to the antibody, because a decrease in the background signal and increase in the sensitivity of the label detection can potentially reduce the antigen detection threshold infinitely [24]. In contrast to the competitive assay, the relationship between

a b c

Antigen Antibody Label

Indirect labeling module

Fig. 3. Immunoassay formats with indirect labeling: a) competi- a b c tive assay with labeled antibody; b) competitive assay with labeled antigen; c) sandwich assay. Antigen with one epitope Antigen with several epitopes

Antigen–carrier conjugate Antibody Label the analyte concentration and the registered signal (the amount of the bound label) in the sandwich assay is direct. Fig. 1. Main immunoassay formats (see the text for description). This approach can also be implemented in three different ways by changing the sequence of interactions between the antigen and the antibody: (i) simultaneous incubation of the antigen with the labeled and unlabeled antibodies (Fig. 2c); (ii) interaction of the antigen first with the labeled antibody and then with the unlabeled antibody (Fig. 2f); (iii) interaction of the antigen first with the unlabeled antibody and then with the labeled antibody (Fig. 2i). The sandwich assay can be used for monovalent antigens as well. Such efforts were summarized in a recent a b c review [25]. However, antibody binding to the antibody complex with a low-molecular-weight compound requires development of special reagents, which is time- consuming work and significantly limits the use of this assay format, even that in some cases, it exhibits very high sensitivity [26]. Each assay scheme described above can be subdivided into two groups based on the method for label incorporation: (i) direct labeling of the antibody or the antigen d e f via formation of a chemical covalent bond or by adsorption; (ii) label in incorporated into additional modules that interact (in a separate reaction or at one of the assay stages) with the assay components, e.g., anti-species antibodies or high-affinity pairs of reagents, such as biotin–(strept)avidin and barnase–barstar (Fig. 3). Based on the way the label is register after the assay is completed, each of the formats mentioned above can sub- dividing into two groups: (i) measuring the label immobi- g h i lized on a support as part of the immune complex or (ii) measuring the unbound label. Fig. 2. Classification of immunoassay formats. The designations Such classification results in 36 (3 × 3 × 2 × 2) poten- are identical to those in Fig. 1. Numbers 1 and 2 indicate the order tially feasible formats of heterogeneous immunoassay that of reagent addition (see the text for the explanation). can be described according to a single scheme.

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1747 The methods of homogeneous assay, with no separa- LABELS USED IN BIOASSAYS tion of the reagents or use of solid supports, form a separate group of immunoassays. In these methods, anti- There are several key periods in the history of bioas- body—antigen binding is detected based on various indi- say development, during which considerable progress has rect parameters: (i) change in the rotation speed of the been achieved due to the appearance of new labels for labeled antigen after its interaction with the antibody detecting specific complexes. For example, in 1960 Yalow (polarization fluoroimmunoassay) [27, 28]; (ii) modula- and Berson proposed the use of the radioactive isotope tion of the marker fluorescence after formation of the 131I to detect immune complexes [53]. The radioim- immune complexes, such as FRET assay, “quenchbod- munoassay (RIA) significantly increased the sensitivity ies”, and “flashbodies” [29-32]; (iii) decrease in the and accuracy of measurements and turned out to be a ver- activity of the enzyme label due to the difficulties in satile approach that could be used for detection of a wide accessing the substrate after formation of the immune variety of compounds. The importance of this invention complexes [33, 34]; (iv) antibody-modulated assembly of was acknowledged by the 1977 Nobel Prize in Physiology oligomeric enzymes resulting in a change in their catalyt- and Medicine awarded to R. S. Yalow. ic activity [35]; (v) changes in the optical, acoustic, and Despite the indisputable advantages of RIA, its other parameters due to the immune complex formation application has been limited due to the complexity of of [36-38], etc. working with isotopes. Therefore, enzymes have been In addition, there are the so-called pseudohomoge- introduced as alternative labels. The methods for anti- neous assay formats in which immunoreagent attached to body labeling with enzymes and registration of enzymat- a carrier, but the antibody–antigen interactions take ic activity had already been partially developed for place in a solution, and only after that, the carrier togeth- immunohistochemical applications. However, numerous er with the bound immune complexes is separated from further advances were required to ensure high yields of other components of the reaction mixture. The most immunoreagent—enzyme conjugates and their stability widely used variant of pseudohomogeneous assay is one [54-57]. As a result, as early as the 1980s, kits and semi- in which the magnetic particles are used as a dispersed automatic and automatic devices for enzyme-linked carrier and then separated using an external magnetic immunosorbent assay (ELISA) became commercially field [39, 40]. Assay systems that use charged polymers available. At about the same time, devices for polarization precipitated by counterions, heat-sensitive polymers, and fluorescence immunoassay (PFIA) were successfully other compounds as carriers have also been described commercialized, and automatic systems for high- [41-44]. throughput testing were introduced [58].. We should also mention bioassays in which antibod- During the 1990s, there was a sharp expansion in the ies are analytes and antigens act as receptor molecules, production of at-home diagnostic test systems based on such as serodiagnosis and identification of specific the immunochromatographic assay (ICA). In these test immunoglobulins (antibodies) in the blood used to diag- systems, the assay has been reduced to the contact nose infections and allergies [45]. In such systems, the between the analyzed sample and the test strip covered antigen is usually immobilized on a solid support, and the with immunoreagents. The movement of the liquid under label is conjugated with the reagents that bind to the influence of capillary forces along the test strip initi- immunoglobulins (anti-species antibodies, protein A ates all the necessary reactions and results in the staining from Staphylococcus aureus, protein G from Streptococcus of certain zones of the test strip. The main markers in spp., or other reagents) [46, 47]. If specific immunoglob- immunochromatography are gold nanoparticles and col- ulins are present in the test sample, detectable complexes ored latex particles [59, 60]. antigen–specific immunoglobulins–labeled immuno- Active interest in nanotechnology at the beginning of globulin-binding protein are formed on the support. the 21st century led to the examination of a wide variety Label-free assay systems, as well as systems that reg- of nanoparticles as a replacement for traditional labels in ister the complex formation from the label incorporation existing assay formats and resulted in the development of in it, have a substantially simpler order of interactions, new bioassays and biosensors. Immunoanalytical applica- the theoretical description of which could be approximat- tions have been suggested for different classes of nanopar- ed by the “antigen + antibody = complex” bimolecular ticles, such as colloidal gold particles of different shapes, reaction. Piezoelectric immunosensors, immunosensors colloidal silver, carbon nanoparticles, magnetic nanopar- based on the registration of surface plasmon resonance, ticles, quantum dots, upconverting fluorophores, infrared and a number of other biosensors are based on this prin- labels, liposomes, silicon nanoparticles, etc. [61, 62]. As a ciple [48-52]. rule, when a new label is proposed for application in Therefore, when considering mathematical models bioassays, it is common to consider the analytical charac- of bioassays, it is necessary to take into account what teristics of the assay that could be achieved with this label detectable complexes are formed and in what order these in comparison to the traditional assay. However, massive complexes are formed in bioassays. methodical solutions are updated relatively slowly. Not

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1748 SOTNIKOV et al. every improvement in assay sensitivity described in the lit- with subsequent amplification of the signal from the erature becomes the basis for the assay modification in oligonucleotide marker by standard polymerase chain practice. reaction (PCR) is particularly interesting. This immuno- The modern diversity of bioassay labels is presented PCR method drastically reduces the immunoassay detec- in the article by Zelenakova et al. [63]: insoluble particles tion limits and can register the minimum number of mol- (erythrocytes, latex granules, graphite particles, sols of ecules in the order of tens or hundreds [67, 68]. metals and dyes, e.g., of gold and Luminous Red G), Supersensitive assay systems based on the detection of radionuclides (125I, 131I, 3H), enzymes (horseradish per- single immune complexes have being actively developed, oxidase, alkaline phosphatase), electron-scattering sub- although at present they require sophisticated instrumen- stances, bacteriophages, fluorescent dyes, chemilumines- tation [69, 70]. cent and bioluminescent compounds, liposomes, pros- thetic groups; enzyme substrates, and enzyme modula- tors. KEY CONCEPTS AND PROCESSES Such diversity stipulates for the criteria that would FOR MODELING BIOASSAY SYSTEMS aid in drawing conclusions about the competitive potential of a new label before or even instead of its experimen- When modeling a bioassay, a number of factors tal evaluation. These criteria have to summarize the should be taken into account in addition to the assay intrinsic characteristics of the label without being related scheme: (i) whether equilibrium or non-equilibrium con- to a particular assay format or analyzed compound, so ditions of interaction are considered; (ii) whether reac- that the label could be appropriately described before tions for forming specific complexes are considered being used in the development of an assay system, there- reversible or irreversible; (iii) whether diffusion-depend- by significantly reducing the labor cost of the studies. ent processes are taken into account; (iv) whether the In our opinion, the following parameters should be model reflects the possibility of bi- and polyvalent inter- taken into account when evaluating a label: actions in the analyte–receptor complexes, etc. 1. Detection limit of the label (per certain medium Although any bioassay is based on the interaction of volume or per certain area of the support). an analyzed compound (analyte) with a biomolecule 2. The maximum number of label units (molecules) specifically binding this analyte (receptor) and it models that can be attached to the immunoreagent without should describe the kinetic properties of this interaction, impairing its immunochemical properties. the above questions could be answered differently, 3. The ratio between the detected signals from the depending on the assay conditions. Accordingly, different label in free and bound forms (the attenuation of the sig- theoretical apparatuses could be chosen to describe a nal in the assay system should be taken into considera- bioassay. tion). Quantitative description of diffusion. As noted above, These three parameters determine the coefficients in most modern bioassays, detectable immune complexes for recalculating the label characteristics for a particular are formed on a solid support. Therefore, theoretical assay system, and therefore, they are very important for description of these assays cannot be limited to solving quantitative descriptions. relatively simple problems of interaction in a solution and However, a number of other important criteria exist should accommodate the diffusion of one of the reactants that are related to the practical aspects of the label appli- toward the support on which another reactant is immobi- cation and can prevent the use of otherwise promising lized and the corresponding inhomogeneity in the distri- labels. These criteria include [60]: bution of reagent concentrations in the reaction volume. 1. The cost of the label and its availability. Consequently, the proposed model should include either 2. Simplicity of the label conjugation with description of the diffusion-dependent processes in immunoreagents. explicit form or justified recommendations for the use of 3. Label stability during storage and reproducibility certain simplifying assumptions that would take into of registered signals. account specific features of this particular assay system. 4. Possibility of label quantification with available Diffusion-limited reactions are typical for heteroge- and affordable (serially manufactured) instrumentation. neous systems [71] and/or systems where the interaction 5. Lack of the influence from the components of the takes place in a viscous medium, e.g., precipitation reac- tested samples on the label signal. tion in a gel [3]. However, only a few of proposed bioassay Modern bioassays are not limited to the use of labels mathematical models consider diffusion processes. The with low detection limits. Many assays involve additional general questions of diffusion-dependent processes in amplification processes that increase the number of label assays were discussed in the classical papers of Stenberg molecules (particles) bound to the immune complex or and Nygren [72-74], but, with a few exceptions [75-77], used for the signal registration [64-66]. For example, have not been elaborated further. Most authors of the assay integration of antibody-based detection of an analyte mathematical models introduce conditional concentra-

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1749 tions in a volume for immobilized reagents and then dis- which makes it possible to use the one-dimensional form cuss the classical patterns of homogeneous reactions. of Eq. (1). This approximation is completely justified, The change in the concentration of the diffusing considering small transverse sizes (5-15 µm) of pores in component (C) over time at the point with coordinates (x, the membranes used in immunochromatographic assays. y, z) is determined by Fick’s second law: In recent years, numerous papers have been published that deal with various aspects of diffusion-dependent processes in bioassays, such as effects of rotational (1) diffusion on the antigen transport [81], the use of fractal models to describe microheterogeneous processes on the The value of diffusion coefficient D determines the biosensor surfaces [82], the choice of the geometry of amount of material passing through a unit of surface per binding microzones to minimize the diffusion-dependent unit of time [78, 79]. The value of D is measured in m2/s processes [83, 84], the design of microfluidic systems with and depends on the properties of both the solvent and the rapid achievement of the chemical equilibrium [85], the solute. In the approximation for the Brownian spherical elimination of diffusion in pseudohomogeneous systems particle, the diffusion coefficient is expressed as: using nanoparticles [86] and polymers [87], etc. Unlike flow-through bioassay systems, including (2) immunochromatographic tests, most other assay formats employ regimes in which reagents have time to redistrib- ute in the volume. When the reagents are incubated for where k is the Boltzmann constant, T is the temperature tens of minutes in milliliter or submilliliter volumes, dif- of the medium, µ is the viscosity coefficient, and r is the fusion can be neglected. In assays with shorter incubation particle radius. times, special procedures are used to accelerate diffusion In assay modeling, Fick’s second law is used not only in the near-surface layer, such as mixing, ultrasound to describe the rate of reactant movement to the receptor treatment, etc. [88-93]. surface [71] but also to characterize the reactant distribu- Quantitative description of chemical transformations. tion in the flow in flow-through systems. For example, The quantitative description of chemical transformations Qian and Bau [80] used Fick’s second law to account for in assay systems not limited by diffusion is based on the the diffusion of reactants in the stream of a liquid sample main postulate of chemical kinetics, i.e., the law of mass in the mathematical description of immunochromatogra- action. For the analyte–receptor reaction in its simplest phy in the sandwich format. According to the Qian and case of monovalent binding (A + R ↔ AR), the law of Bau’s model, the change in the label (colloidal gold) con- mass action provides the differential equation for the rates centration at a point with the coordinate x (the direction of changes in the reagent and complex concentrations of the x axis coincides with the direction of the fluid flow) [94]: is described as follows:

(4) (3)

where ka and kd are kinetic association and dissociation where A is the detected compound (analyte), P is the ana- constants of the AR complex, respectively. lyte-binding sites on the label, R is the receptor in the Taking into account that [A] + [AR] = [A]0 and capture zone for analyte binding, PA is the analyte com- [R] + [AR] = [R]0, where the subscript 0 denotes initial plex with the label, RPA is the analyte complex with the concentrations of the reagents, Eq. (4) could be reduced label and the receptor in the capture zone, x is the coor- to the form with a single variable [AR]: dinate of the position on the test strip, Fn is the rate of the n-th complex formation, U is the fluid flow rate, and D is p . (5) the label diffusion coefficient. The expression U·d[concentration of reagent]/dx in Eq. (3) reflects the change in the concentration of P at a Equation (5) has a rigorous analytical solution, point with coordinate x resulting from the fluid flow that which is the function: shifts positions of all dissolved molecules and particles. 2 2 The expression Dp·d [concentration of reagent]/dx reflects the change in the concentration of P at a point (6) with coordinate x resulting from the diffusion processes described by Fick’s second law. Qian and Bau’s model assumes that the properties of the assay system do not differ in the y and z directions, where a = 2[A]0[R]0, b = [A]0 + [R]0 + kd/ka.

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1750 SOTNIKOV et al. This exact solution is almost never used in the mod- B cell receptor (about 8.5 min). Further increase in the els described in published literature; instead, approximate half-life of the complex no longer promotes proliferation solutions are applied. When the kinetics of affinity com- of B cells. Therefore, during the secondary immune plexes formation is described, two main approximations response, the values of the equilibrium association con- are adopted as a rule: (i) equilibrium conditions of the stant for the anti-antigen immunoglobulins G (IgG) are interaction; (ii) the irreversibility of the binding reaction. within the range of 107-1010 M–1. The equilibrium state is defined by the condition that The limitations on the selection of high-affinity all reaction rates equal zero, from which follows the clones described above do not exclude the possibility that expression: for some antigens, a significantly higher degree of com- plementarity with the antigen-binding site in the antibody and a correspondingly higher association constant can be (7) achieved. Non-dissociating antigen–antibody complexes were described in [104]. In this study, antibodies with an where Ka is the equilibrium association constant, or the infinite affinity for the chelate complex of (S)-benzyl- affinity constant. EDTA with indium were obtained as a result of a directed This is a very rough approximation for express bioas- antibody design. Similar effects were observed for the says, in which reagents are incubated for several minutes interaction between the antigen and the antibody–metal only, although it is used in practice [80, 95, 96]. When ion complex in [105]. characterizing kinetic systems, the approximation that Experimental evaluation of interactions parameters in the binding reaction is irreversible is better justified, as the bioassay. Although finding the analytical solution in a receptor molecules used in bioassays usually exhibit high general form for a system of equations is the best way of affinity for the detected compound. For example, anti- theoretically describing a bioassay, characterization of a bodies rarely have kinetic dissociation constant greater bioassay requires the knowledge of quantitative parame- than 10–4 s–1 [97], i.e., less than 6% of the formed ters of the analyte–receptor interactions, first of all, immune complex would dissociate within 10 min. kinetic and equilibrium reaction constants. Knowing Therefore, dissociation in the kinetic assay system can be these parameters allows to evaluate the validity of neglected. assumptions used in the model development and, if nec- The product formation in the bimolecular irre- essary, to resort to numerical modeling. versible reaction A + R → AR is described by Eq. (8) [98]: The methods for measuring association and dissociation constants of biomolecular complexes are very diverse [106]. For example, equilibrium dialysis and elu- (8) tion of reagents from affinity columns have been widely used for this purpose [107]. A theoretical apparatus for Note that Eq. (8) is the limiting case of Eq. (6) for the calculation of association constants in capillary elec- kd → 0. trophoresis experiments has been developed [108-110]. Theoretical limits of parameters of the antigen–anti- The methods based on fluorescence and fluorescence body reaction. The traditionally mentioned advantage of polarization measurements enabled direct rapid charac- antibodies is a high affinity of antibody–antigen complex terization of the ligand–receptor interactions in a solu- that allows detection of antigens in extremely low con- tion [111-113]. ELISA is another popular approach to centrations. However, the affinity of the immune complex measuring the constants, although it also represents a is limited by the nature of the immune response induc- type of bioassay that is constantly developed, as well as tion, which imposes limitations on the detection limit of used for characterization of reagents in the development immunoassays. of other assays. Classical recommendations on this sub-

The maximum association rate constant (ka) for the ject have been proposed Friquet and Djavadi-Ohaniance interaction of antibodies with protein antigens is deter- et al. in [114-116]. The details of processing of experi- mined by the rate of diffusion of the protein molecules in mental data and the choice of the most valid procedures solution; it is of the order of 106 M–1·s–1 [99]. The disso- for calculations are discussed in a number of later works ciation rate constant of the complex (kd) can be increased [117-121]. by the antibody selection in vivo [100, 101], but only to a Despite all this diversity, generally accepted gold certain limit. According to the affinity ceiling hypothesis standard for measuring and comparing the constants of [101-103], further antigen-driven selection of clones intermolecular interactions is the biosensor assay based already secreting high-affinity antibodies occurs less effi- on the registration of surface plasmon resonance (SPR). –4 –1 ciently. This is because for antibodies with kd = 10 s , Let us consider in detail the principle of the operation the half-life of the antigen complex with the B cell recep- and data processing for the Biacore biosensor system, as tor is almost 2 h, which is several times greater than the the most widely used among optical sensor systems duration of the endocytosis of antibody complex with the [122].

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1751 The experimental procedure for the intermolecular interaction characterization in the Biacore system includes: Signal upon binding 1) immobilization of one of the interacting components on the chip surface; 2) passing a solution containing the second compo- Regeneration nent of the interacting pair over the chip surface, with Sample injection Signal (RU) simultaneous registration of the interaction; Baseline 3) ceasing the inflow of the second component into Buffer Buffer Buffer the cell and registering the dissociation of the formed complex; Time, s 4) regeneration of the chip surface with solutions that disturb the ligand–receptor complex but do not Fig. 4. General form of the sensogram registered in the Biacore damage the immobilized component; system. RU, resonance units. 5) repeating stages 2 through 4 with other prepara- tions that react with the immobilized component. The dependence of the device response (the SPR value) on time is called the sensogram [123]. Its principal (10) general form is shown in Fig. 4. The sensogram contains all the data necessary to cal- RU culate kinetic and equilibrium parameters of the complex RU (11) formation [124-127]. The simplest model that allows to RUmax RUmax calculate the constants from the sensogram assumes the equilibrium conditions and excludes diffusion-dependent Equation (11) describes the signal on the equilibrium processes from consideration [125-127]. association constant and the concentration of the analyte Association stage. Taking into account the equilibri- entering the cell. Assuming the equilibrium state, RU val- um approximation, the association stage is described by ues closest to the equilibrium values (the upper plateau of Eq. (4), where the time derivatives of concentrations the association part of the sensogram) should be substi- equal zero: tuted into the equation. The interaction constant could be calculated using solutions with known concentrations (9) of the analyte. Dissociation stage. Dissociation of the immune com- Since the concentration of active receptor molecules plex is a monomolecular reaction, and its rate depends on the sensor surface ([R]) is usually unknown, it is more only on the kinetic dissociation constant. In a flow- convenient to consider the degree of surface filling (θ) through system, dissociation can be considered irre- instead. The θ value is the RU/RUmax ratio, where RU is versible, since dissociation products are washed away the signal value (minus the background) and RUmax is the from the reaction zone, and new complexes cannot form maximum signal value (minus the background) when all because of the analyte absence in the washing fluid. the receptor molecules are bound to the surface. The Therefore, the dependence of the dissociated complex concentrations of free and bound receptor molecules can content (D) on the time of dissociation (t) is determined be expressed through the initial concentration of the by the kinetic equation of the irreversible monomolecular receptor [R]0 and θ: [R] = (1 – θ)[R]0 and [AR] = θ [R]0 reaction: [128]. Because at the association stage, the analyte con- (12) stantly enters the reaction zone with the fluid flow and, according to the assumptions, is immediately uniformly The portion of the dissociated complex is calculated mixed, its concentration above the sensor surface can be as D = 1 – RU/RU0, where RU0 is the signal value considered constant, i.e. [A] ≈ [A]0. The increase in the (minus the background) at the moment when dissociation analyte concentration near the surface resulting from its begins, and RU is the signal value (minus the back- specific binding can also be neglected, as the analyte ground) at the moment of time t. amount at the sensor surface is much smaller than in the Bivalence of antibodies and its role in immunoassay. volume; therefore, we can assume that the average analyte An important feature of antibodies that distinguishes concentration due to the interaction with the receptor them from other receptor molecules is the presence of varies insignificantly. The higher the fluid flow rate, the multiple binding sites for the analyte. Immunoglobulins more accurate this approximation would be. Thus, Eq. G, that are most commonly used in assays, have two anti- (9) can be rewritten as follows: gen-binding sites, whereas immunoglobulins M have 10

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1752 SOTNIKOV et al. binding sites1. If identical epitopes on the antigen surface the so-called nodal analysis (NODE). This method con- are located at a sufficient distance from each other, this siders the kinetics of each reaction of a multicomponent antigen can bind not one but two antibodies. Such poly- process in a separate node within a short time interval and valent antigens are viral particles, bacterial cells, biopoly- then constructs the iterative process that encompasses the mers with degenerate structure, some oligomeric pro- whole reaction period. The model could be made more teins, and synthetic antigen–carrier conjugates. complex by introducing additional reactions and, accord- The bivalent antibody–antigen interaction results in ingly, the nodes for the calculations. a number of consequences important for the bioassay In general, the authors of studies on bivalent evaluation: (i) formation of a mixture of immune com- immune interactions note the need for considering the plexes of different composition and affinity between anti- differences between the affinity of a simple interaction bodies and polyvalent antigens; (ii) significantly lesser between the antigen epitope and the antibody binding and probability of dissociation of the antigen–antibody com- affinity of an average immune complex that reflects poly- plexes with the bivalent binding. and monovalent interactions for a given pair of reagents. In competitive immunoassays, polyvalent interac- The ratio between the equilibrium association con- tions of antibodies with a standard antigen (usually a syn- stants for bi- and monovalent antibody–antigen interac- thetic analyte–carrier conjugate) impair the detection tions can be up to two orders of magnitude [146, 147]. It sensitivity, because complexes with a monovalent antigen depends on the antibody and the antigen used and from the analyzed sample are less stable than those with a reflects the mechanism of interaction between the anti- polyvalent antigen. Therefore, the “worsening” of the body active site and the corresponding antigen epitope, standard antigen by reducing the analyte content in it the mobility of the immunoglobulin molecule segments, would increase the assay sensitivity [129-131]. and the distance between antigenic determinants and Various aspects of the polyvalent interactions in their orientation on the surface of the polyvalent antigen. bioassays have been discussed in [132-140]. Shiau [141] Crothers and Metzger [148] calculated association con- analyzed the immune complex formation in a system constants for the two active sites of the antibody based on sisting of two types of monoclonal antibodies and a biva- experimentally determined values of the monovalent lent antigen with two identical epitopes. The author did interaction. not consider the possibility of the cyclic complex formation, as well as of the cooperativity effect, and studied only the average molecular weight of linear antigen–anti- MODELS OF VARIOUS BIOASSAYS body complexes. Joshi [142] used the statistical-mechan- ical theory to model the dependence of the formation of Competitive immunoassay. The principle of the clas- complexes of different composition on the antibody affin- sical competitive immunoassay with a labeled antigen ity. Joshi considered two systems: interactions between (Fig. 1b) is a simultaneous interaction of antibodies (Ab) monoclonal antibodies and an antigen of a known valen- with an antigen (Ag) and its labeled analog (Ag*) [149]. cy with identical determinants, and a more complicated Accordingly, there are two reactions in the system. case where the properties of the binding sites of the antibodies and the antigens differed. A theoretical study on (1) Ab + Ag ↔ AbAg the effect of clusterization of receptors in various config- urations on the kinetic parameters of immune interac- (2) Ab + Ag* ↔ AbAg* tions was presented by van Opheusden et al. [143]. Hlavacek et al. [144] proposed a model for the multivalent The reactions are characterized by the kinetic asso- ligand–receptor binding, that took into account the ciation constants ka1 and ka2 and kinetic dissociation con- shielding of several receptors by a ligand (typical for virus- stants kd1 and kd2. In the simplest case, all binding sites in es and multivalent protein antigens) and studied the effect the antibodies are assumed to be identical. All the trans- of this factor on the kinetic and equilibrium parameters of formations in this system are described by the scheme the reaction. The properties of assay systems using anti- shown in Fig. 5. gen-loaded liposomes were examined by Hendrickson et Let us construct the calibration curve for this system, al. [145]. i.e., the dependence of the detected signal (the amount of An attempt to take into account polyvalent interactions without overcomplicating calculations techniques was made by the authors of [11], who used in their model

1 Fragmented antibodies from certain animals (the so-called nanobodies), recombinant antibodies with only one antigen- binding site, and specially prepared antibody fragments are Fig. 5. Reactions in the competitive immunoassay with a labeled monovalent. antigen (see the text for explanations).

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1753 bound label) on the analyte concentration in the tested equal to the content of the free unlabeled antigen, sample. The examination of this function allows theoret- because ical investigation of the influence of various parameters on the analytical characteristics of the method. Based on the law of mass action, we can write a system of differential equations: (24)

where x is the proportion of the free antigen (labeled or (13) unlabeled). If the proportions of the free labeled and unlabeled antigens are equal, then the proportions of the bound (14) labeled and unlabeled antigens are equal as well:

(25) (15) Adding Eqs. (19) and (20), after the transformations, The system should be supplemented with equations we obtain: expressing the law of conservation of mass: (26) (16)

Considering that [AbAg] = y[Ag]0; [AbAg*] = (17) y[Ag*]0; [Ag] = (1 – y)[Ag]0; [Ag*] = (1 – y)[Ag*]0; [Ab] = [Ab]0 – ([AbAg] + [AbAg*]) = [Ab]0 – y([Ag]0 + (18) [Ag*]0), Eq. (26) can be rewritten as:

Here and below, the index 0 denotes the initial con- (27) centrations of the reactants. Note that [Ab] is the total concentration of the binding sites in the antibodies, Equation (27) can be found in the literature in vari- because antibodies are polyvalent. ous forms; for example, in [150], the equation is trans- Several variants of the analytical solution from the formed to: system of Eqs. (13)-(18) have been described in published literature. As a rule, the approximation of equilibrium (28) conditions is used [2, 149] that allows to convert Eqs. (13)-(15) in a system of algebraic equations: In fact, this is a quadratic equation that unites the proportion of the bound antigen, the initial concentra- (19) tions of the binding sites of the antibodies, the concentrations of the labeled and unlabeled antigen, and the inter- (20) action constant (complex dissociation constant):

Expressing Eqs. (14) and (15) through the equilibrium association constants results in: (29) (21) The solution of the quadratic Eq. (29) describes in general form the assay calibration curve: (22)

For simplicity, in the first competitive assay models the affinities of the antibodies to the antigen and its (30) labeled analog were assumed to be identical, i.e., Ka1 = Ka2 = Ka [2, 150]. In this case: Later, the competitive assay models were considered (23) for the case when the antibody affinities to the labeled and unlabeled antigens differ [22, 151, 152]. Such system Expression (23) means that if the constants are is described by the cubic equation derived from Eqs. (16)- equal, then the content of the free labeled antigen is (22):

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(31) that simultaneously interacts with the immobilized antigen and the free antigen in the solution (competitor) (Fig. where: 2a). However, the kinetic model presented above is equal- ly suitable for describing this variant of the assay (impor- a = Kd1 + Kd2 + [Ag]0 + [Ag*]0 – [Ab]0, tantly, in both systems the diffusion of the reagents to the b = Kd2([Ag]0 – [Ab]0) + Kd1([Ag*]0 – [Ab]0) + Kd2Kd1, surface is not considered), except that some of the desig- c = –Kd2Kd1 [Ab]0. nations must be changed: Ab is the labeled antibodies, Using the trigonometric formulas, the solution of and Ag* is the immobilized antigen. The detected com- Eq. (31) is: plex is AbAg*, as in the first case. Another competitive assay format is the sequential (32) saturation method, in which the antibodies first interact with the antigen to be detected (reaction (1)), and then, where after washing-off, with the labeled antigen (reaction (2)). Rodbard et al. [153] showed the that this method is much more sensitive than the procedure involving simultaneous incubation of the antibodies with the unlabeled and Having found [Ab], we can write the formulas for labeled antigens. Based on the analysis of the mathemat- [AbAg] and [AbAg*]: ical model of this system, Dzantiev and Yuriev formulat- ed recommendations for improving the assay characteristics [154]. In the sequential saturation method, the assay (33) parameters are determined by the reaction with the antigen to be detected. The reaction with the labeled antigen plays an auxiliary role by allowing the number of free binding sites to be registered. When the strict proportion- (34) ality [AbAg*] = C*[Ab] is achieved, the calibration curve in the normalized coordinates should completely coin- cide with the antigen titration curve. In reality, this pro- An example of the equilibrium concentrations of portionality might be violated; therefore, assay optimiza- components in a competitive assay system that were cal- tion should be selection of conditions under which the culated from Eqs. (32)-(34) while keeping in mind Eqs. calibration curve corresponds best to the titration curve. (16)-(18) is shown in Fig. 6. Of particular interest is the The authors of [154] also showed that in order to fulfill dependence of [AbAg*] on the concentration of the this requirement, the incubation time with the labeled added antigen, because it is essentially a calibration curve antigen should equal to approximately 0.1/kd1. A good of the assay. Analysis of this curve will allow evaluation of correlation between the experimental and calculated data potential analytical characteristics of the method and the has been shown for the insulin ELISA as an example. influence of the initial parameters on them. If in order to characterize the extent of reaction with In the alternative competitive assay format, the anti- a detected antigen, we introduce the coefficient n (the gen is immobilized, and the labeled reagent is an antibody proportion of the antibody binding sites filled with the antigen after equilibrium is established), then the following formulas are valid:

n[Ab]0 = [AbAg] = [Ab]0 – [Ab], n = 1 – [Ab]/[Ab]0.

ations, µM Using the expressions for the equilibrium association

constant, [AbAg] = Ka1[Ab][Ag] = Ka1([Ab]0 – [AbAg])([Ag]0 – [AbAg]), we can write:

n[Ab]0 = Ka1([Ab]0 – n[Ab]0)([Ag]0 – n[Ab]0). (35)

From here, we can obtain an expression that relates

Equilibrium concentr the proportion of the filled binding sites in antibodies to log (concentration of added antigen) the association constant and the initial concentrations of the reagents: Fig. 6. Calculated equilibrium concentrations of components in a competitive assay system. Parameters: [Ab]0 = 2 µM; [Ag*]0 = (36) –1 –1 1 µM; Ka1 = 5 µM ; Ka2 = 10 µM .

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1755 Equations (30), (34), and (36) describe the calibration curves of various competitive assay formats, predict the behavior of systems with varying concentrations of reagents and association constants, and establish theoretical limits for detection by this method, ranges of detectable concentrations and other assay parameters (see section “Theoretical evaluations of the potential of bioassay systems”). Sandwich immunoassay. In the sandwich immunoassay, detected antigen has two epitopes for binding two antibodies: one of these antibodies is immobilized on the support, while the other antibody is labeled. Ab1 is the first antibody; Ab2* is the labeled second antibody, and Ag is the detected antigen. The classical sandwich assays with the immobilization of the first antibody (RIA and

ELISA) involve seven steps: 1) immobilization of the Ab1 antibody; 2) washing off unbound antibodies; 3) incuba- Fig. 7. Reactions in the sandwich immunoassay (see the text for tion with the antigen Ag, i.e., Ab1 + Ag ↔ Ab1Ag (1A) explanations). reaction with the association constant k1 and dissociation constant k2; 4) washing off unbound Ag; 5) incubation with the labeled Ab*2 antibody, i.e., Ab1Ag + Ab*2 ↔ (40) Ab1AgAb*2 (1B) reaction with the association constant k3 and dissociation constant k4; 6) washing off unbound (41) labeled antibodies; 7) measurement of the signal proportional to the concentration of the [Ab1AgAb*]2 complex. (42) Ideally, only Ab1 molecules and Ab1Ag complexes remain on the surface after stage 4, after which the Ab1Ag In total, a system of six equations with six unknowns, complex dissociates with the formation of free antigen [Ag], [Ab1], [Ab*],2 [Ab1Ag], [AgAb*],2 and [Ab1AgAb*],2 molecules. Therefore, two additional reactions take place was obtained; p, q1, and q2 are the total concentrations of during stage 5: Ab*2 + Ag ↔ Ab*Ag2 (1C) and Ab1 + the bound and free Ag, Ab1, and Ab2, respectively. AgAb*2 ↔ Ab1AgAb*2 (1D) with the association constants k5 The authors solved this system with numerical meth- and k7 and dissociation constants k6 and k8, respectively. ods or analytically with the approximation of equilibrium The complete scheme of the reactions in the sand- conditions [155]. The analytical solution after substitu- wich immunoassay is shown in Fig. 7. tion of R = [Ab1Ag]/[Ag] after reaction (1A) provides the Rodbard and Feldman [155] analyzed these reac- following equation: tions based on the assumption that the first antibody binds to the support irreversibly, and the unbound (43) reagents are completely washed away at each stage. The reactions in the sandwich immunoassay could Taking into account tall the accepted assumptions, be described by a system of differential equations: the initial conditions for the following reactions are determined by the equations:

(44) (37) (45)

(46) (38) The solution to Eq. (43) with respect to R gives the following expression:

(39) (47)

Three additional algebraic equations follow from the After R is found, the concentrations of the remaining law of conservation of mass: components of the system could be calculated. Figure 8

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1756 SOTNIKOV et al. shows the results of numerical modeling as dependences duration [171, 172]. Most attention has been given to the of the concentrations of different products of interaction calculation of the theoretical limits of analyte detection at the last stage of the assay. in various immunoassay formats [173-176]. The presented model shows that when the concen- Because the analyte–receptor complexes are detect- tration of the added analyte increases, the concentration ed in one way or another in all biochemical immunoassay, of the detectable complex [Ab1AgAb*]2 should increase the equilibrium and kinetic constants of analyte–receptor monotonically. However, some experimental data binding directly affect the assay parameters. However, the demonstrate that at high analyte concentrations, the analyte detection limit is also influenced by the sensitivi- dependence can be reversed [156]. This phenomenon is ty of the label detection, experimental errors, and non- called the hook effect [157, 158]. Because of it, Rodbard specific interactions in the system. and Feldman modified their model and showed that the The detection limit and the quantitative measure- hook effect occurs when subpopulations of antibodies ment limit are frequently confused, as it was pointed out with different affinities are present or when the antigen is by C. F. Woolley et al. [23]. The detection limit is the not completely washed away after the reaction with anti- amount of the analyte that gives a signal that reliably bodies [159]. exceeds the background, whereas the quantitative meas- Other mechanisms underlying the hook effect have urement limit corresponds to the amount of the analyte been established as well. For example, the hook effect can that is measured with a certain accuracy, which the be observed when labeled antibody competes with the researcher considers sufficient. Therefore, the concept of unlabeled one for the binding sites on the antigen, but the detection limit has a qualitative character and reflects only if the concentration of labeled antibody is lower than only the presence of the analyte in a sample. These issues the concentration of the unlabeled antibody [160]. The are especially important in the context of rapidly develop- hook effect has also been described for the one-stage ing methods for a single molecule measurement, which sandwich assay, in which labeled and unlabeled antibod- have become possible due to the appearance of ultrasen- ies react with the antigen simultaneously (Fig. 2c) [161]. sitive measuring systems that can detect the activity of a Zherdev and Dzantiev showed in a microplate ELISA, single label molecule (particle) [177-179]. In practice, that the hook effect can result from the desorption of the detection of a single molecule in a sample means only a first antibody during the subsequent assay stages. They high probability of the presence of this molecule in a sam- also demonstrated that in order to minimize the influence ple. The limit of quantitative measurements in such sys- of desorption and dissociation effects, it is important to tems is much higher than single molecules. carry out incubation with the second antibody as quickly Let us consider in more detail the theoretical evalu- as possible and not longer than for 1/kd [162]. ations of the limiting analytical characteristics of various There are a number of additional approaches for bioassays and their relationship with the parameters of bioassay modeling and theoretical investigation of the affinity interaction. influence of experimental conditions on the assay charac- Jackson–Ekins criterion. The theoretical detection teristics. Among the promising mathematical description limits for noncompetitive and competitive immunoassays techniques, fractal analysis is of particular interest. Sadana methods and their relationship with antibodies character- et al. used fractal analysis for examining different types of istics were the first time established by Jackson and Ekins bioassays in [163-165]. Theoretical evaluation of the tem- [176]. perature influence on the ELISA characteristics was pre- They showed that in competitive immunoassays, sented by Muller in [166]. Software tools for optimizing experimental errors and antibody affinity determine the competitive ELISA were provided by Sittampalam et al. in detection limit, as expressed by the following ratio: [167]; similar approaches were developed by Tsoi et al. in

[168]. Theoretical approaches to selecting conditions for σmin,0 = CV0 /Ka, (48) expanding the working range of ELISA were discussed by

Ohmura et al. [169]. Model and Healy reviewed the meth- where σmin,0 is the theoretical minimum of the standard ods for optimization of assay conditions with specific deviation of the system response at zero analyte concen- attention to the procedure sensitivity and cost in [170]. tration, and CV0 is the variation coefficient of the system response at zero analyte concentration. Jackson and Ekins also found that for immunoassays THEORETICAL EVALUATION with a direct correlation between the analyte concentra- OF THE BIOASSAY POTENTIAL tion and the detected signal value, the detection limit can potentially be reduced by several orders of magnitude. Development of the bioassay model provides the However, for noncompetitive methods, nonspecific inter- possibility for the theoretical evaluation of the limiting actions in the system can also significantly influence the analytical characteristics of the method. The main bioas- detection limit in addition to experimental errors and say parameters are the detection limit, specificity, and antibody affinity, which is described by the expression:

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1757

σmin,0 = Kn*CV/Ka, (49) where σmin,0 is the theoretical minimum of the standard deviation of the system response at zero analyte concentration, CV is the relative error of the system response at zero analyte concentration, and Kn is the relative value of nonspecific binding. For example, if the antibody association constant is 1012 M–1, CV is 1%, and the level of nonspecific binding is 1%, the theoretical minimum of the standard deviation is 10–16 M. Expressions (48) and (49) were derived based on the assumption that the label has an unlimited activity and Equilibrium concentration does not restrict the detection limit. Taylor et al. [150] used the Jackson–Ekins model to calculate the detection limits for different Ka and CV0 values in the competitive assay using the standard require- Fig. 8. Calculated equilibrium concentrations of components in ment that the minimum detected signal should be three the sandwich assay. Parameters: q1 = 3/K1 = 0.3; q2 = 0.5; k1 = times the standard deviation of the baseline signal (table). k7 = 5; k3 = k5 = 10; k2 = k4 = k6 = k8 = 1 (according to [155]). As follows from the data presented, it is theoretically possible to achieve a detection limit in the femtomolar concentration range, if the antibodies used have with the Antigen (A) affinity constant of 1012 M–1. Based on the Jackson– Labeled antibodies (P) Liquid front Ekins model, it can be assumed that at the same antibody affinity, the detection limit of the noncompetitive assay Movement of liquid sample might be lower than the detection limit of the competitive Immobilized antibodies (R) method by a value that is inversely proportional to the nonspecific binding level (Kn). This implies a possibility of a potentially unlimited lowering of the detection limit with decreasing Kn. In practice, if the nonspecific binding is eliminated, the detection limit will be limited by the label detection sensitivity. The model by Jackson and Ekins significantly influenced the development of bioassays by defining the boundaries of the methods that use receptors and mark- Liquid sample Analytical zone ers. Although a number of authors discuss how to over- Reaction: Reactions: come the limitations on the assay sensitivity imposed by this model, it is not a question of refuting it, but rather developing new approaches to detecting immune complexes, including methods for a single molecule detection [23, 177, 180-182].

Fig. 9. A simplified scheme of the sandwich ICA used for the Theoretical detection limits (DL) for competitive model development. immunoassay, calculated according to the model by Jackson and Ekins in [150] Relationship between the parameters of affinity inter- K (M−1) DL ( = 3%) DL (CV0 = 1%) CV0 actions and assay characteristics. The process for opti- 1012 30 fM 90 fM mization of a biochemical assay, which includes multiple 1010 3 pM 9 pM components and stages, usually requires a large number of laborious experiments [167, 183]. To simplify this 2·109 15 pM 45 pM process, a number of optimization schemes have been 8 10 300 pM 900 pM developed, such as the Doehlert matrix [184], the Box- 107 3 nM 9 nM Behnken design [185], the Taguchi design [186, 187], and 106 30 nM 90 nM other [168, 188, 189]. However, these schemes do not 105 300 nM 900 nM allow an a priori assessment of maximum achievable assay parameters.

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1758 SOTNIKOV et al. There are very few publications on the correlation the differences in the absorption in different microplate between the affinity constants and assay characteristics. wells (σW): Rodbard and Lewald [190] were the first to propose an algorithm for constructing the RIA calibration curves in (53) order to predict the detection limits. The theoretical model of RIA proposed by Ezan et al. [191] based on the law of mass action gives the dependence: where A is the analyte concentration, and G is IC50. Based on the quantitative parameters of the affinity complex formation, the proposed model makes it possible (50) to establish the optimal conditions for competitive immunoassays. where C is the unlabeled antigen concentration, K is the Dzantiev and Yuriev analyzed the relationship antibody affinity, [Ag*] is the initial concentration of the between the parameters of affinity interaction and the labeled antigen, b0 is the ratio of the antigen–antibody characteristics of competitive assays with sequential satu- complex concentration to the total antigen concentration ration [146]. They showed that if [Ab]0 < 1/Ka1, then the in the absence of unlabeled antigen, and bc is the same detection limit is determined mainly by the value of ratio in the presence of unlabeled antigen. 1/Ka1; and if [Ab]0 > 1/Ka1, then the detection limit is Within the framework of this model, Ezan et al. determined by the value of [Ab]0. Therefore, the use of showed the decisive effect of the parameter bc on the antibodies with an initial concentration of ~1/Ka1 would achievable detection limit: be optimal. An increase in the antibody concentration

from two orders of magnitude (0.1/K – 10/K) at [Ab]0 << 1/K , to one order of magnitude (0.1[Ab] – [Ab] ) at (51) a1 0 0 [Ab]0 >> 1/Ka1 narrows the dynamic range of the detected concentrations. However, when the antibody concen- where t is the Student’s criterion, s is the standard devia- tration decreases, the relative error in antigen determination of the parameter b0, n is the number of measurements tion increases due to the decrease of the calibration curve for one antigen concentration, n0 is the number of deter- slope in the dynamic range area. minations of parameter b0, and bDL is the parameter bc value for the unlabeled antigen concentration corresponding to the detection limit. FLOW-THROUGH BIOASSAYS

The actual detection limit (CDL), taking into account Eq. (50), is calculated by the formula: Coming after traditional immunoassay methods, such as RIA, ELISA, etc., flow-through bioassays have now established their place in the laboratory practice. (52) However, open heterogeneous flow-through require tran- sition to the models of a much greater complexity. A This expression shows how to vary the assay condi- number of works have been devoted to the theoretical tions in order to achieve the minimum detection limit. description of flow-through biosensors [195-199]. The For example, when working with high-affinity antibodies proposed models take into account the reaction system 10 –1 (Ka > 10 M ), the concentration of the labeled antigen geometry and the effect of diffusion transport on the should be reduced. If the antibody affinity is much lower, kinetics of binding of dissolved ligands to the immobilized then the amount of the labeled antigen should be receptors [200]. Despite the complexity of the flow- increased to reduce the measurement error. through heterogeneous systems, models have been devel- Similar correlations between the affinity constants oped based on the minimum number of assumptions that and assay characteristics have been analyzed for other provide general analytical description of the interactions methods, such as the RIA [181], capillary immunoelec- in these systems and can be used for predicting the prop- trophoresis [150], ELISA [192, 193], and immunochro- erties of their functioning. matography [80, 95]. Below, we present the results of mathematical mod- The influence of the measurement accuracy on the eling of ICA systems, which are the most popular now immunoassay characteristics was discussed in [194]. among the lateral-flow immunoassays due to the simplic- Theoretical evaluation of the immunoassay detection ity of their use and interpretation of the results. limits and working ranges was presented [192]. The model An important feature of the membrane-based sys- by Hayashi et al. [194] takes into account the relative tems is the difference in the properties of immunore- standard deviations (ρT) for the following parameters: agents in solution and immobilized immunoreagents. The errors for pipetting antigen (ρA), antigen–enzyme conju- use of effective constants determined for one type of the gate (ρG), antibody (ρB), and enzyme substrate (ρS) and assay in an alternative assay is rather conventional; the

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 MATHEMATICAL MODELING OF BIOASSAYS 1759 same is true for the amount of immobilized immunore- (57) agents in terms of the volume concentration. In 2003-2004, Qian and Bau developed for the first time the analytical (non-numerical) models for the ICA in the sandwich and competitive formats [80, 95]. A sim- (58) plified scheme for the sandwich ICA used for its modeling is shown in Fig. 9. When the liquid front moves along (59) the test strip membrane, the antigen first interacts with the labeled antibody. After the liquid front reaches the capture zone, the antibody–antigen complex interacts (60) with the secondary antibody immobilized in the capture zone, resulting in the formation of the detected triple complex. , (61) To account for the kinetics of the complex formation, diffusion, and liquid flow in the test strip, a system where kai is the kinetic association constant of the i-th of equations is introduced: reaction, and kdi is the kinetic dissociation constant of the i-th reaction. (54) This system of equations cannot be solved in the general form. Therefore, the authors made the assumption on the equilibrium conditions, which allowed to calculate (55) the concentration of the RPA complex in the capture zone (which is proportional to the color development intensity) according to the following equations: , (56) (62) where A is the detectable compound (analyte), P is the sites for analyte binding on the label, R is the receptor in the capture zone for analyte binding, PA is the analyte complex with the label, RA is the analyte complex with the receptor, RPA is the label–analyte–receptor com- (63) plex in the capture zone, x is the coordinate of the position on the test strip (the system is considered one- The index e denotes the equilibrium concentration, and dimensional), Fn is the rate of the n-th complex forma- the index 0 denotes the initial concentration. tion, U is the liquid flow rate, and Dp is the diffusion The same authors proposed the mathematical mod- coefficient. els for description of competitive ICA. The models have Expressions in a form of U·d[reagent concentra- been developed for two cases: tion]/dx reflect the change in the concentration of the 1) The analyte binds to the antibody immobilized on corresponding reagent (A, P, or PA) at a point with coor- the membrane, thereby blocking the interaction of the dinate x due to the liquid flow, while expressions in a form immobilized antibodies with the labeled conjugate. The 2 2 of Dp·d [reagent concentration]/dx reflect the change in higher the amount of the analyte, the fewer free binding the concentration of the corresponding reagent (A, P, or sites on the membrane for the conjugate, and the lower PA) at a point with coordinate x due to diffusion. the signal level (Fig. 2b). Complexes containing R are formed only in the capture 2) The analyte binds to the conjugate of the labeled zone; therefore, their concentration does not depend on antibodies and interferes with the interaction of the con- the flow and diffusion. jugate with the antigen molecules immobilized on the The accepted assumption [80, 95, 201] that the dis- membrane. The higher the amount of the analyte, the tribution of components in a flow obeys the Fick’s second fewer binding sites on the conjugate that are available for law does not take into account a number of factors, such binding in the capture zone (Fig. 2d). as nonspecific interactions with membranes, desorption, Remember that this dependence was derived under the possibility of preliminary mixing of reagents, etc. the assumption of the equilibrium conditions of Therefore, by introducing the parameters that reflect dif- immunochromatography. However, according to data by fusion processes, Qian and Bau [95] actually analyzed a Ragavendar and Anmol [96], at parameters typical for –8 –8 6 simplified version of the equations for reagents uniformly real immunoreagents (A0 = 10 ; P0 = 10 ; ka = 10 ; kd = distributed in the volume. 10–3) and a fluid flow rate of 0.5 mm/s, the reactions on Qian and Bau further analyzed a system of equations the test strip will approach the equilibrium only if the describing the formation of AP, AR, and APR complexes: capture zone is located about 12 cm from the position of

BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017 1760 SOTNIKOV et al. the sample application. This distance significantly ical model for such system. The authors showed that in exceeds the size of the working membrane of a standard order to lower the detection limit for specific antibodies immunochromatographic test (2.5 cm). Therefore, in in ICA, it is necessary to use the highest possible reagent reality, immunochromatographic reactions take place concentrations for antibody binding and to dilute the under nonequilibrium conditions and require a more sample at least ten times before performing the assay. complex mathematical description. In all the above-mentioned works, the conjugates of The published models of ICA that take into account the labels with receptor molecules were presented as a set nonequilibrium processes use numerical approaches to of analyte-binding sites distributed in a reaction volume. calculate the kinetics of the immune interactions [202- These models do not take into account the dimensional 204]. parameters of the label particles and the composition of To describe the immune complex formation in the their conjugates. An attempt to evaluate the influence of capture and control zones of the test strip, these parameters on the characteristics of the sandwich Krishnamoorthy et al. [205] proposed a model that takes ICA was undertaken by Liu et al. in a recent study [206]. into account diffusion in membrane pores in combination According to their conclusions, conjugates that have with a system of differential equations that reflect the about 30 analyte-binding sites on one label particle are interactions in solution. The model views the flow as a best suited for the use in sandwich ICA. two-phase mixture: a liquid phase that partially or completely fills the pores and air that occupies the remaining The presented review of the published literature con- space. The authors of the paper assumed that the flow is firms that there is still a need for theoretical description of two-dimensional, and there is no loss of reagents from the bioassays. Although it is impossible to measure all quanti- membrane surface due to evaporation. If the volume is tative parameters of the reagents used in each known averaged, the flow density can be regarded constant, and assay or to predict theoretically the value of the detected the dissipation due to the viscosity can be neglected. The signal, general descriptions of bioassays (primarily based position of the liquid front is proportional to the diffusion on numerical solutions describing their functioning in a constant, which depends on the pore size, viscosity, and general form) are in high demand. Having established the surface tension. This system has been numerically charac- principles of the system behavior, it is possible to formu- terized for different variants of reagent flow (water, col- late general recommendations for choosing reagent con- loidal gold in water, and colloidal gold conjugate with BSA centrations and duration of the assay stages, to identify in water) across membranes with different pore sizes (from factors that limit the maximum sensitivity, and to adapt 4 to 20 µm). The model describes the influence of mem- assay systems for solving specific practical problems. brane porosity and the position of the binding lines on the Theoretical models can determine the full potential of number of immune complexes that form. The results of various assay formats, evaluate the effectiveness of the experimental and theoretical studies [205] show that the proposed solutions for signal amplification, and shorten flow rate in an immunochromatographic system decreases the duration of the interaction in an assay. Using the the- with the distance from the starting line, i.e., position of the oretical approach will make the process of assay develop- binding zone significantly affects the sensitivity of the ment less labor-consuming and allow comparison of dif- assay. The flow rate moderately increases with the increas- ferent assays. ing pore size. This reduces the incubation time and, correspondingly, the intensity of the registered staining. These correlations were confirmed by Berli and Kler in Acknowledgments their analysis of the sandwich ICA model [15]. Hence, the sensitivity of ICA is influenced by time, concentration of The authors thank A. N. Berlina (Centre for reagents, pore size, and position of the capture zone. Biotechnology, Russian Academy of Sciences) for the ICA for determination of specific antibodies has preparation of Figs. 1-3. many features that require separate theoretical consider- This work was supported by the Russian Science ation. 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BIOCHEMISTRY (Moscow) Vol. 82 No. 13 2017